Over the years, it turned out to be possible to characterise thoroughly the \(  \texttt {S}  \) -matrices for more involved quantum field theories underlying to other
integrable classical field theories in 1+1 dimensions. Such models possess several types of asymptotic particles which can also form bound states.
Then, the "in" Fock Hilbert space is more complicated and takes the form \( \bigoplus \limits _{n=0}^{+\infty } L^2(\mathbb {R}^n_{>},\otimes ^n \mathbb {C}^{p})\)  where
the \(L^2\) -space refers to \(\otimes ^n \mathbb {C}^{p}\)  valued functions on \(\mathbb {R}^n_{>}\) , with \(p\)  corresponding to the number of different asymptotic particles in the given theory.
The most celebrated example corresponds to the Sine-Gordon quantum field theory. Building on Faddeev-Korepin's [1]} semi-classical quantisation results of the solitons
in the classical Sine-Gordon model, one concludes that the underlying quantum field theory possesses two distinct types of asymptotic
particles of equal mass, the soliton and the anti-soliton, as well as a certain number, which depends on the coupling constant, of bound states thereof.
These all have distinct masses and are called breathers. Zamolodchikov argued the explicit form of the \(  \texttt {S}  \) -matrix governing the soliton-antisoliton scattering [2]}
upon using the factorisability of the \(n\) -particle \(  \texttt {S}  \) -matrix into two-particle processes,
the independence of the order in which a three particle scattering process arises from a concatenation of two-particle processes as well as
the fact that equal mass particles may solely exchange their momenta during scattering, this due to the existence of many conservation laws.
This enforces that the \(  \texttt {S}  \)  matrix
satisfies the Yang-Baxter equation, which originally appeared in rather different contexts
[3]}, [4]}, and strongly restricts its form.
We do stress that the Yang-Baxter equation is the actual cornerstone of quantum integrability, so that it is not astonishing to recover it also in this setting.
The missing pieces of the Sine-Gordon \(  \texttt {S}  \) -matrix capturing the soliton-breather and breather-breather scarrering were then proposed in [5]}.
Nowadays, \(  \texttt {S}  \) -matrices of many other models have been proposed, see e.g. [6]}, [7]}.
